Overall, Brouwer fixed point theorem and Kakutani fixed theorem are non-constructive. Is there any established paper that demonstrates that there exists constructive proofs that do exactly what these theorems do?
Yes, there are constructive proofs for fixed point theorems including Brouwer. Also, the proof of the Banach fixed point theorem with which I am most familiar is constructive. In fact, here is a paper all about constructive methods for fixed point theorems: